What is the angle between `vec(A)` and the esultant of `(vec(A) + vec(B))` and `(vec(A) - vec(B))`

Options

A. `0^(@)`
B. `tan^(-1)((A)/(B))`
C. `tan^(-1)((B)/(A))`
D. `tan^(-1)((A-B)/(A+B))`

Answer

To find the angle between the vector \(\vec{A}\) and the resultant of the vectors \((\vec{A} + \vec{B})\) and \((\vec{A} - \vec{B})\), we can follow these steps:

### Step 1: Define the Resultant Vector
We need to calculate the resultant vector \(\vec{R}\) which is given by:
\[ \vec{R} = (\vec{A} + \vec{B}) + (\vec{A} - \vec{B}) \]

### Step 2: Simplify the Resultant Vector
Now, simplify the expression for \(\vec{R}\):
\[ \vec{R} = \vec{A} + \vec{B} + \vec{A} - \vec{B} \]
The \(\vec{B}\) terms cancel out:
\[ \vec{R} = 2\vec{A} \]

### Step 3: Determine the Angle Between \(\vec{A}\) and \(\vec{R}\)
Now, we need to find the angle \(\theta\) between \(\vec{A}\) and \(\vec{R}\):
\[ \vec{R} = 2\vec{A} \]
Since \(\vec{R}\) is simply \(\vec{A}\) scaled by a factor of 2, both vectors point in the same direction.

### Step 4: Conclusion on the Angle
The angle \(\theta\) between two vectors that point in the same direction is:
\[ \theta = 0^\circ \]

Thus, the angle between \(\vec{A}\) and the resultant of \((\vec{A} + \vec{B})\) and \((\vec{A} - \vec{B})\) is \(0^\circ\).

### Final Answer
The angle is \(0^\circ\).

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